To apply the fictitious domain method and conduct numerical experiments, a boundary value problem for an ordinary differential equation is considered. The results of numerical calculations for different values of the iterative parameter τ and the small parameter ε are presented. A study of the auxiliary problem of the fictitious domain method for Navier-Stokes equations with continuation into a fictitious subdomain by higher coefficients with a small parameter is carried out. A generalized solution of the auxiliary problem of the fictitious domain method with continuation by higher coefficients with a small parameter is determined. After all the above mathematical studies, a computational algorithm has been developed for the numerical solution of the problem. Two methods were used to solve the problem numerically. The first variant is the fictitious domain method associated with the modification of nonlinear terms in a fictitious subdomain. The model problem shows the effectiveness of using such a modification. The proposed version of the method is used to solve two problems at once that arise while numerically solving systems of Navier-Stokes equations: the problem of a curved boundary of an arbitrary domain and the problem of absence of a boundary condition for pressure in physical formulation of the internal flow problem. The main advantage of this method is its universality in development of computer programs. The second method used calculation on a uniform grid inside the area. When numerically implementing the solution on a uniform grid inside the domain, using this method it’s possible to accurately take into account the boundaries of the curved domain and ensure the accuracy of the value of the function at the boundaries of the domain. Methodical calculations were carried out, the results of numerical calculations were obtained. When conducting numerical experiments in both cases, quantitative and qualitative indicators of numerical results coincide.

At the beginnig we will give a literature review on the application of the fictitious domain method. Currently, there are several methods for the numerical solution of boundary value problems in complex geometric domains, such as the method of curved grids and the fictitious domain method. The construction of curved grids for the numerical solution of problems requires the transformation of the equation into curved coordinates, which has a more complex form than the original equations. When constructing curved grids, various requirements are imposed on difference equations, which makes the construction of curved grids a difficult mathematical task. Therefore, for the numerical solution of a wide class of problems of mathematical physics in an arbitrary domain, it is effective to use the fictitious domain method [

The fictitious domain method for the Navier-Stokes equations is the subject of works by Bugrov et al. [

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The fictitious domain method is utilized in solving problems of computational fluid dynamics such as modeling the motion of particles in a fluid flow [

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Glowinski et al. [

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Let us dwell on the advantages and disadvantages of the existing variants of the fictitious domain methods. The method proposed and implemented in [

Therefore, in this paper, the authors propose a method that allows constructing a homogeneous difference scheme in the entire extended domain which is a convenient tool in terms of programming automation. At the same time, a reasonable continuation of the main equation coefficients leads to the convergence of the solution to the desired solution in the original domain, which is confirmed by mathematically proven statements and the results of numerical calculations.

In this work, two methods are applied for the numerical solution of the formulated problem. The first one is the fictitious domain method associated with the modification of nonlinear terms in a fictitious subdomain. The model problem shows the efficiency of using such a modification. The proposed variation of the method is used to solve two problems at once that arise in the numerical solution of the Navier-Stokes equations: the problem of the curvilinear boundary of an arbitrary domain and the problem of the absence of a boundary condition for pressure in the physical formulation of the internal flow problem.

The second method used the calculation on a uniform mesh inside the domain. The solution on a uniform mesh inside the domain makes it possible to accurately take into account the boundaries of the curved domain and ensures the accuracy of the function value on the boundaries of the domain in the numerical implementation.

Consider the boundary value problem for the ordinary differential equation:

We apply the fictitious domain method for problem

The coefficients in

If

If

By comparing the solutions of the one-dimensional model problem, we see that in the first case, when

We consider the following non-stationary equation for the numerical implementation of the one-dimensional equation of the fictitious domain method by the establishment method:

The coefficients are defined as follows:

Let us construct a non-uniform mesh that thickens in the neighborhood of the actual boundary,

The steps of a non-uniform mesh thickening near

Then the corresponding difference scheme on a nonuniform mesh has the form

Iteration parameter, |
0.001 | 0.0001 | 0.0001 | 0.0001 |
---|---|---|---|---|

Small parameter, |
10^{−3} |
10^{−6} |
10^{−9} |
10^{−12} |

Non-uniform steps of the mesh | 0.01 | 0.01 | 0.01 | 0.01 |

Error, |
0.00003329 | 0.00003343 | 0.00003343 | 0.00003343 |

Number of iterations, |
338 | 336 | 336 | 336 |

Number of iterations, |
338 | 336 | 336 | 336 |

The auxiliary problem corresponding to the fictitious domain method is reduced to solving a system of nonlinear equations with variable coefficients in

Condition

Let us introduce a set of infinitely differentiable solenoidal in

The following lemma [

To apply the Galerkin method, consider the eigenvalue problem

The operator

In matters related to the justification of the existence of solutions to boundary value problems for the Navier-Stokes equations by the Galerkin method, the very fact of the existence of the spectral problem eigenfunctions for the operator

The operator is not symmetric and positive definite in the considered problem

Assume that

Since

Taking into account

Further,

The solvability of systems of

Global solvability follows from a priori estimates of the solution

By virtue of the continuity equation and boundary conditions, we have

Hence the following inequality is obtained:

Let

Then

Choose

Then it follows from inequality

We consider the numerical solution of the auxiliary problem

Let us take a curvilinear channel with solid boundaries (

We use the finite difference method and the scheme of splitting by physical processes [

Let the velocity field

Stage I:

Stage II:

Stage III:

The prescribed values of pressure and zero values of the tangential component of the fluid flow velocity at the inlet and outlet of the computational domain were set in the numerical implementation. At the «solid» boundaries, the pressure values are given as linear functions and the tangential component of the velocity is equal to zero.

Obviously, the sum of the equations corresponding to Stages I and III gives the original equation of motion

The following physical interpretation of the given splitting scheme is proposed. At Stage I, it is assumed that the transfer of momentum (momentum per unit mass) is carried out only due to convection and diffusion. The velocity field thus obtained does not satisfy the incompressibility condition, in general. In this paper, the implicit scheme is used at Stage I in contrast to the classical version of the method of splitting into physical processes [

At Stage II, the pressure field is found from the solution of the Poisson equation based on the found intermediate velocity field and taking into account the solenoidality condition of the velocity vector. At this stage, numerical methods for solving grid elliptic equations with Dirichlet boundary conditions for pressure were used.

At Stage III, it is assumed that the transfer is carried out only due to the pressure gradient, and the convection and diffusion are absent.

In the numerical implementation, grids containing

The upper and lower solid curved boundaries are described by the equations

Currently, there are several methods for the numerical solution of boundary value problems in complex geometric domains, such as the method of curved grids and fictitious domain method. The construction of curved grids for the numerical solution of problems requires the transformation of the equation into curved coordinates, which has a more complex form than the original equations. Therefore, for the numerical solution of a wide class of problems of mathematical physics in an arbitrary domain, it is effective to use the fictitious domain method.

The solution of this problem is implemented in two ways. In the first case, the fictitious domain method by the leading coefficients was used. The main advantage of this method is its versatility in the development of computer programs for the numerical simulation of a wide class of problems of mathematical physics. In the second case, a calculation on a was used. Methodological calculations have been carried out, and the results of numerical calculations have been obtained. The quantitative and qualitative indicators of the numerical results coincide for both cases when conducting numerical experiments.

The boundary of the physical domain is smeared when solving the problem by the fictitious domain method, and therefore the solutions at the boundaries may differ from the boundary condition, although it gives reliable results of the flow. In addition, the fictitious domain method is easily implemented. But since the problem is ill-conditioned at the first stage, an implicit scheme was used where the boundary conditions of the integer iteration step were used in the calculations.

Numerical implementation of the solution on a uniform grid inside the domain makes it possible to accurately take into account the boundaries of the curved domain and ensures the accuracy of the function value on the domain boundaries. The only drawback of this approach is the lack of a specific algorithm, and one has to come up with separate conditions for determining the boundary and boundary nodes of the grid for each problem, especially when boundary conditions of the second kind are imposed.

Sections | X = 0.5 | X = 1.0 | X = 1.5 | |||
---|---|---|---|---|---|---|

Velocity components | U | V | U | V | U | V |

The values of velocity components obtained by the method of fictitious areas | 0.563424339 | 0.127813428 | 0.573329524 | 0.000405213 | 0.523948447 | 0.050608793 |

The values of velocity components obtained by the consistent grid method | 0.543891 | 0.107817 | 0.561694 | 0.000262 | 0.494597 | 0.107749 |

Difference of the values | 0.019533 | 0.019996 | 0.011635 | 0.000144 | 0.029351 | 0.05714 |

It can be seen from

In

The calculations used a uniform grid with dimensions of

Thus, in order to apply the fictitious domain method and conduct numerical experiments, the boundary value problem for an ordinary differential equation is first considered. The results of numerical calculations for different values of the iterative parameter τ and the small parameter ε are presented. After successful application of the fictitious domain method for an ordinary differential equation, a more complex problem of applying the fictitious domain method for the Navier-Stokes equation in natural variables is considered.

Further, the research of the auxiliary FDM problem at the differential level for the Navier-Stokes equations with continuation into a fictitious subdomain by the higher coefficients with a small parameter is carried out. Methods of a priori estimates are used for the mathematical study of the problems under consideration. A generalized solution of the auxiliary FDM problem with continuation by higher coefficients with a small parameter is determined. The fictitious domain method is used to solve many problems of computational fluid dynamics. Currently, there are several methods for the numerical solution of boundary value problems in complex geometric domains, such as the method of curved grids and fictitious domain method. The construction of curved grids for the numerical solution of problems requires the transformation of the equation into curved coordinates, which has a more complex form than the original equations. Therefore, for the numerical solution of a wide class of problems of mathematical physics in an arbitrary domain, it is effective to use the fictitious domain method.

Thus, in this paper, two methods are applied for the numerical solution of the formulated problem. The first one is the fictitious domain method associated with the modification of nonlinear terms in a fictitious subdomain. The model problem shows the efficiency of using such a modification. The proposed variation of the method is used to solve two problems at once that arise in the numerical solution of the Navier-Stokes equations: the problem of the curvilinear boundary of an arbitrary domain and the problem of the absence of a boundary condition for pressure in the physical formulation of the internal flow problem. The main advantage of this method is its versatility in the development of computer programs.

The second method used the calculation on a uniform mesh inside the domain. The solution on a uniform mesh inside the domain makes it possible to accurately take into account the boundaries of the curved domain and ensures the accuracy of the function value on the boundaries of the domain in the numerical implementation.

Number of mesh nodes | 50 × 20 | 100 × 40 | 150 × 60 | 200 × 80 |
---|---|---|---|---|

Methods | ||||

Fictitious domain method | 2.14 s. | 7.03 s. | 15.37 s. | 1 min., 39.36 s |

Consistent grid method | 9.01 s. | 25.43 s. | 1 min., 7.15s. | 1 min., 33.25 s |

Number of mesh nodes | 50 × 20 | 100 × 40 | 150 × 60 | 200 × 80 |
---|---|---|---|---|

Methods | ||||

Fictitious domain method | 1987 | 2172 | 2258 | 2329 |

Consistent grid method | 2184 | 1999 | 2363 | 2607 |

Authors thank those who contributed to write this article and give some valuable comments.